Overshooting by differential heating

TL;DR

This paper investigates the process of differential heating in stars, deriving scaling relations and using simulations to show how weak flows can cause mixing in stellar radiative zones, especially near convection boundaries.

Contribution

It introduces a simplified model of differential heating, derives new scaling laws, and confirms them with high-resolution simulations relevant to stellar mixing processes.

Findings

Flow speed drops abruptly at a finite height above the heat source.

Differential heating induces mixing extending about 4% of the pressure scale height.

Numerical simulations confirm the analytic scaling relations.

Abstract

On the long nuclear time scale of stellar main-sequence evolution, even weak mixing processes can become relevant for redistributing chemical species in a star. We investigate a process of "differential heating," which occurs when a temperature fluctuation propagates by radiative diffusion from the boundary of a convection zone into the adjacent radiative zone. The resulting perturbation of the hydrostatic equilibrium causes a flow that extends some distance from the convection zone. We study a simplified differential-heating problem with a static temperature fluctuation imposed on a solid boundary. The astrophysically relevant limit of a high Reynolds number and a low P\'eclet number (high thermal diffusivity) turns out to be interestingly non-intuitive. We derive a set of scaling relations for the stationary differential heating flow. A numerical method adapted to a high dynamic range in flow amplitude needed to detect weak flows is presented. Our two-dimensional simulations show that the flow reaches a stationary state and confirm the analytic scaling relations. These imply that the flow speed drops abruptly to a negligible value at a finite height above the source of heating. We approximate the mixing rate due to the differential heating flow in a star by a height-dependent diffusion coefficient and show that this mixing extends about $4\%$ of the pressure scale height above the convective core of a $10\,M_\odot$ zero-age main sequence star.